Earlier today I set two puzzles involving prime numbers – in particular 3, 7 and 11. Primes are on this month’s cultural radar thanks to the Apple TV+ thriller Prime Target, which stars hunk du jour Leo Woodall (above.)
Here they are again with solutions.
1. Prime power
Without using a calculator, what is the last digit of 3 to the power of a million, i.e 31,000,000?
Solution: 1
When you multiply 3 by itself, the final digit of the answer cycles through the digits 3, 9, 7 and 1. Since a million is divisible by four, the final digit of 3 to the million is 1.
2. The 7-Eleven
A shopper walks into a convenience store and buys four items. The cashier says the total is £7.11.
‘How funny,’ says the shopper. ‘That’s the name of the shop.’
‘What’s also funny is that I multiplied the prices of the four items together,’ says the cashier.
‘Aren’t you supposed to add the items together?’ replies the shopper.
‘You can do if you like – the total is exactly the same.’
What are the prices of the items?
Solution £3.16, £1.25, £1.20 and £1.50
Let the prices be a, b, c and d. Then we know
[1] a + b + c + d = 7.11
[2] a x b x c x d = abcd = 7.11
We can assume that the prices in the shop are in pounds and pence, so lets multiply each of them by 100 to get rid of the decimal point.
[3] 100a + 100b + 100c + 100d = 711
[4] 100a x 100b x 100c x 100d = 100,000,000abcd = 711,000,000
To keep things neat, let A = 100a, B = 100b, etc, so we can rephrase the above.
[3] A + B + C + D = 711
[4] ABCD = 711,000,000
We need to solve these two equations, and we do this by first finding the prime factors of 711 million. (In the post where I set the puzzle I explained that all numbers can be broken down into their constituent prime numbers.)
The prime factorisation of 711 million is 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5 x 5 x 5 x 5 x 5 x 79. ( I said you were allowed to use this site to find prime factors)
Now we work out which of these prime factors make up A, B, C and D.
You do this through trial and error, which can be time consuming so at noon I gave the hint that £3.16 was a solution. This corresponds to 316 (since we multiplied the prices by 100). The prime factors of 316 are 79 x 2 x 2. So lets say A = 79 x 2 x 2.
The hint also said that the other three prices divide by 5p. Thus we know that 5 is a factor of B, and D. We also know from [3] that B + C + D = 395.
By playing around with the prime numbers in our factorisation, we get B = 5 x 5 x 5 = 125, C = 5 x 3 x 2 x 2 = 120, and D = 5 x 5 x 3 x 2 = 150. Thus the prices are £1.25, £1.20 and £1.50. Check them if you dont believe me! They add up and multiply to £7.11.
What’s great about this puzzle not so much the solving of it, but the fact that it is amusing to find such an arithmetical gem in a global brand.
I hope you enjoyed the puzzles, I’ll be back in two weeks.
Thanks to Steve Thompson who suggested the first puzzle, and to the PR team at Apple TV+.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
